Implementing the maximum likelihood methodology to measure a driver’s critical gap

Most of the capacity calculation procedures for two-way stop-controlled (TWSC) intersections are based on gap acceptance models. Critical gap is one of the major parameters for gap acceptance models. The accuracy of capacity estimation is mainly determined by the accuracy of the critical gap. This paper focuses on the implementation of the maximum likelihood technique to measure a driver’s critical gap using field data. A methodology to define gap events is proposed, so that the accepted gaps and maximum rejected gaps required by the maximum likelihood technique could be obtained. Specific issues regarding multi-lane situations and major street right turn movement are discussed. Special conditions observed during the research are addressed when the proposed method cannot be applied directly, such as the existence of a mid-block refuge area where minor street drivers can seek gaps in a two-stage process, pedestrian blockage, and downstream queue spill back. The proposed method was adopted in measuring critical gap under US conditions during a research project, described by Kyte et al. (1996). ©     

Estimation of gap acceptance parameters within and across the population from direct roadside observation

This paper explores the feasibility of maximum likelihood as an approach to determine the parameters of gap acceptance functions when these functions vary from individual to individual. Specifically, it is shown that it is theoretically possible to estimate the average critical gap of a population of drivers (or pedestrians) and its variance, within and across individuals, from direct roadside observations. Although the Multinomial Probit Model provides a natural theoretical framework for the estimation of these parameters, the model seems not to be statistically estimable for this particular problem. It was shown, however, that if one of the parameters is known, the other two become estimable and a two-stage estimation process that takes into account this phenomenon can be utilized. The technique is demonstrated with the 203-driver data set included in Appendix A. The Multinomial Probit Model can also be used to determine simultaneously the mean critical gap, the mean critical lag (the first gap considered by a driver), and the variances of these. For the data set in Appendix A, the mean critical gap was significantly smaller than the mean critical lag, as one might expect. The techniques proposed in this paper have the further advantage of being statistically efficient with large data sets and of not requiring a panel of individuals to be observed under controlled conditions.     

Estimation of an integrated driving behavior model

This paper presents the methodology and results of estimation of an integrated driving behavior model that attempts to integrate various driving decisions. The model explains lane changing and acceleration decisions jointly and so, captures inter-dependencies between these behaviors and represents drivers’ planning capabilities. It introduces new models that capture drivers’ choice of a target gap that they intend to use in order to change lanes, and acceleration models that capture drivers’ behavior to facilitate the completion of a desired lane change using the target gap.The parameters of all components of the model are estimated simultaneously with the maximum likelihood method and using detailed vehicle trajectory data collected in a freeway section in Arlington, Virginia. The estimation results are presented and discussed in detail.     

Estimation of multinomial probit-kernel integrated choice and latent variable model: comparison on one sequential and two simultaneous approaches

Integrated choice and latent variable (ICLV) model incorporates latent factors into standard discrete choice model with aim to provide greater explanatory power. Using simulated datasets, this study makes a comparison among three estimation approaches corresponding to the sequential approach and two simultaneous approaches including the maximum simulated likelihood with GHK estimator and maximum approximate composite marginal likelihood (MACML) approach, to evaluate their abilities to recover the underlying parameters of multinomial probit-kernel ICLV model. The results show that both simultaneous approaches outperform the sequential approach in terms of estimates accuracy and efficiency irrespective of the sample sizes, and the MACML approach is the most preferable due to its best performance on recovering true values of parameters with relatively small standard errors, especially when the sample size is large enough.     

Linear probit models: Statistical properties and improved estimation methods

The multinomial probit model is a statistical tool that is well suited to analyze some transportation problems. Modal split, gap acceptance, and route choice are some examples of application contexts. This paper presents an in-depth analysis of its statistical properties and an estimation method for the trinomial case. In the statistical part of the paper it is shown that for multinomial probit models with specifications that are linear in the parameters, the global maximum of the log-likelihood function is consistent if the data do not exhibit multicollinearity as defined in the text. For the special case with three alternatives, lack of multicollinearity is also shown to guarantee asymptotic efficiency and normality, and the uniqueness of any root of the likelihood equations. In addition, it is also shown that for the trinomial probit model certain goodness-of-it measures and test statistics can be easily calculated. The methods part of the paper introduces an estimation process that solves the likelihood equations using a special purpose table of the bivariate normal distribution and analytical derivatives of the log-likelihood function. The method is very accurate, can be applied to nonlinear specifications, and is considerably faster than current computer programs. For linear specifications, the method can be mathematically proven to converge if the log-likelihood equations have a root.     

Maneuvers of motorcycles in queues at signalized intersections

The maneuvering models of motorcycles in previous studies often considered motorcycles' traveling in terms of movements in a physical static lane and not in terms of dynamic virtual lane‐based movements. For that reason, these models are not able to imitate motorcyclists' behavior well. This paper proposes a maneuverability model framework for motorcycles in queues at signalized intersections with considering the dynamic motorcycle's lane. The model includes (i) a dynamic motorcycle's lane to identify the current, left, and right lanes of the subject motorcycle, (ii) a threshold distance to determine when a motorcyclist starts to consider maneuvering, (iii) a lane selection model to identify the lane preferred by a motorcyclist, and (iv) a gap acceptance model to describe whether or not the lead and lag gaps are acceptable for maneuvering. The model framework captures the variation across the motorcyclist population and over time observations. The models were applied to Hanoi and Hochiminh city, Vietnam, based on microscopic data collected from video images. All of the parameters were estimated using the maximum likelihood method with the statistical estimation software GAUSS. The results show that 77.88% of the observed maneuvers – either staying in the current lane or turning left or right – could be modeled correctly by the proposed models. Copyright © 2010 John Wiley & Sons, Ltd.     

An adaptive time gap car-following model

We intend to define a continuous car-following model exclusively based on the time gap. A model of the interaction between a vehicle and its predecessor is produced by adjusting the time gap to a targeted safety time that is a function of speed. The model is defined by a differential system, to which a consistent numerical scheme is associated. The parameters of the model are statistically estimated by maximum likelihood. In order to reproduce a heterogeneous traffic flow, vehicles are differentiated by type, and to recreate asymmetric longitudinal behavior, acceleration phases are distinguished from deceleration phases. Introducing a reaction time, inducing a delay in the perception and processing of information about vehicles in interaction, can alter the stability of the flux through appearance of kinematic waves. By simulation, the types and domains of parameters which are asymptotically unstable are identified. The results reveal that, in the model, the statistically estimated parameters form is strongly factor of instability.     

Application of social force model to pedestrian behavior analysis at signalized crosswalk

Limited pedestrian behavior models shed light on the case at signalized crosswalk, where pedestrian behavior is characterized by group or individual evasion with surrounding pedestrians, collision avoidance with conflicting vehicles, and response to signal control and crosswalk boundary. This study fills this gap by developing a microscopic simulation model for pedestrian behavior analysis at signalized intersection. The social force theory has been employed and adjusted for this purpose. The parameters, including measurable and non-measurable ones, are either directly estimated based on observed dataset or indirectly derived by maximum likelihood estimation. Last, the model performance was confirmed in light of individual trajectory comparison between estimation and observation, passing position distribution at several cross-sections, collision avoidance behavior with conflicting vehicles, and lane-formation phenomenon. The simulation results also concluded that the model enables to visually represent pedestrian crossing behavior as in the real world.     

Novel Methods and the Maximum Likelihood Estimation Technique for Estimating Traffic Critical Gap

Most unsignalised intersection capacity calculation procedures are based on gap acceptance models. Accuracy of critical gap estimation affects accuracy of capacity and delay estimation. Several methods have been published to estimate drivers' sample mean critical gap, the Maximum Likelihood Estimation (MLE) technique regarded as the most accurate. This study assesses three novel methods; Average Central Gap (ACG) method, Strength Weighted Central Gap method (SWCG), and Mode Central Gap method (MCG), against MLE for their fidelity in rendering true sample mean critical gaps. A Monte Carlo event based simulation model was used to draw the maximum rejected gap and accepted gap for each of a sample of 300 drivers across 32 simulation runs. Simulation mean critical gap is varied between 3s and 8s, while offered gap rate is varied between 0.05veh/s and 0.55veh/s. This study affirms that MLE provides a close to perfect fit to simulation mean critical gaps across a broad range of conditions. The MCG method also provides an almost perfect fit and has superior computational simplicity and efficiency to the MLE. The SWCG method performs robustly under high flows; however, poorly under low to moderate flows. Further research is recommended using field traffic data, under a variety of minor stream and major stream flow conditions for a variety of minor stream movement types, to compare critical gap estimates using MLE against MCG. Should the MCG method prove as robust as MLE, serious consideration should be given to its adoption to estimate critical gap parameters in guidelines. Copyright © 2012 John Wiley & Sons, Ltd.     

Aggregation with multinomial probit and estimation of disaggregate models with aggregate data: A new methodological approach

This paper describes an analytic aggregation procedure for disaggregate demand models similar to the one proposed in earlier publications by Westin (1974) and McFadden and Reid (1975). The technique, which uses a multivariate normal approximation for the distribution of the vector of attributes, is based on the multinomial profit algorithm proposed by Daganzo, Bouthelier and Sheffi (1977) and can be applied to an arbitrary number of alternatives. The procedure is computationally so efficient that it enables us to calibrate disaggregate models with aggregate data by maximum likelihood using the same or slightly modified codes developed for disaggregated data. The paper also contains a small scale numerical example intended to illustrate the important highlights of the aggregation-estimation problem.     

Estimating “tree” logit models

The existence of the “tree” generalisation of the multinomial logit model, its consistency with theories of “rational” human behaviour, as well as the advantages offered in terms of more realistic modelling by this model form, have all been known for some time. Although a few studies have applied tree logit models, however, applications have been restricted by practical and theoretical difficulties in the estimation of these models, in particular with the sequential estimation that has normally been necessary. This paper presents a new estimation procedure, employing the maximum likelihood criterion, for the estimation of tree logit models. The procedure is based on a simple notation for the tree structure and is believed to be original. It offers the possibility of estimating simultaneously all the parameters of general tree logit structures, without restriction on the number of levels but allowing the user to impose constraints of parameter equality where required. Results are presented indicating both the usefulness of the method in deriving improved models, relative to the more familiar sequential estimations, and its practicality and speed of operation on main-frame, minicomputers and microcomputers.     

A meta-model of vehicle ownership choice parameters

This paper builds a meta-model of vehicle ownership choice parameters to predict how their values might vary across extended periods as a function of macroeconomic variables. Multinomial logit models of vehicle ownership are estimated from repeated cross-sectional data between 1971 and 1996 for large urban centers in Ontario. Three specifications are tested: a varying constants (VC) model where the alternative specific constants are allowed to vary each year; a varying scales (VS) model where the scale parameter varies instead; and a varying scales and constants model. The estimated parameters are then regressed on macroeconomic variables (e.g., employment rate, gas prices, etc.). The regressions yield good fit and statistically significant results, suggesting that changes in the macroeconomic environment influence household decision making over time, and that macroeconomic information could potentially help predict how model parameters evolve. This implies that the common assumption of holding parameters constant across forecast horizons could potentially be relaxed. Furthermore, using a separate validation dataset, the predictive power of the VC and VS models outperform conventional approaches providing further evidence that pooling data from multiple periods could also produce more robust models.     

Convergence of the Frank-Wolfe method for certain bounded variable traffic assignment problems

In two recent papers published in Transportation Research, Daganzo presented a modification of the Frank-Wolfe algorithm to solve certain link capacitated traffic assignment problems satisfying certain conditions. In order to show convergence of the modified algorithm, the assumption was made that the integral of the volume delay formula for each link tends to infinity as the link flow approaches the link capacity. In this paper we give a Theorem which establishes convergence of the modified algorithm under much weaker conditions. This result is then used to show convergence if the objective function of the assignment model is sufficiently large (not necessarily infinite) when the link flows are at capacity. Thus the modified method is applicable to a broader class of assignment problems. Two numerical examples illustrate (a) when the method converges and when it does not, and (b) that our Theorem provides a weaker condition for convergence of the method.     

The maximum approximate composite marginal likelihood (MACML) estimation of multinomial probit-based unordered response choice models

The likelihood functions of multinomial probit (MNP)-based choice models entail the evaluation of analytically-intractable integrals. As a result, such models are usually estimated using maximum simulated likelihood (MSL) techniques. Unfortunately, for many practical situations, the computational cost to ensure good asymptotic MSL estimator properties can be prohibitive and practically infeasible as the number of dimensions of integration rises. In this paper, we introduce a maximum approximate composite marginal likelihood (MACML) estimation approach for MNP models that can be applied using simple optimization software for likelihood estimation. It also represents a conceptually and pedagogically simpler procedure relative to simulation techniques, and has the advantage of substantial computational time efficiency relative to the MSL approach. The paper provides a “blueprint” for the MACML estimation for a wide variety of MNP models.     

Interval estimation in the calibration of certain trip distribution models

This paper argues for interval, rather than point, estimation when calibrating some variants of the trip distribution “gravity” models. Analytic expressions are derived for the approximate asymptotic covariances of least squares and maximum likelihood estimates of the parameters in the impedance function under a variety of conditions. A comparative numerical example, and an application using migration flows, are also presented.     

A new generalized heterogeneous data model (GHDM) to jointly model mixed types of dependent variables

This paper formulates a generalized heterogeneous data model (GHDM) that jointly handles mixed types of dependent variables—including multiple nominal outcomes, multiple ordinal variables, and multiple count variables, as well as multiple continuous variables—by representing the covariance relationships among them through a reduced number of latent factors. Sufficiency conditions for identification of the GHDM parameters are presented. The maximum approximate composite marginal likelihood (MACML) method is proposed to estimate this jointly mixed model system. This estimation method provides computational time advantages since the dimensionality of integration in the likelihood function is independent of the number of latent factors. The study undertakes a simulation experiment within the virtual context of integrating residential location choice and travel behavior to evaluate the ability of the MACML approach to recover parameters. The simulation results show that the MACML approach effectively recovers underlying parameters, and also that ignoring the multi-dimensional nature of the relationship among mixed types of dependent variables can lead not only to inconsistent parameter estimation, but also have important implications for policy analysis.     

Using gap sequences to estimate gap acceptance functions

Traditional gap acceptance functions have been estimated based on the first gap observed. In this paper we show that the critical gap of drivers is decreasing on the average, as they are waiting for an acceptable gap. Our gap acceptance function is based on a probit model which assumes a normal distribution of gaps across gaps and drivers.     

A simulation evaluation of the maximum approximate composite marginal likelihood (MACML) estimator for mixed multinomial probit models

This paper evaluates the ability of the maximum approximate composite marginal likelihood (MACML) estimation approach to recover parameters from finite samples in mixed cross-sectional and panel multinomial probit models. Comparisons with the maximum simulated likelihood (MSL) estimation approach are also undertaken. The results indicate that the MACML approach recovers parameters much more accurately than the MSL approach in all model structures and covariance specifications. The MACML inference approach also estimates the parameters efficiently, with the asymptotic standard errors being, in general, only a small proportion of the true values. As importantly, the MACML inference approach takes only a very small fraction of the time needed for MSL estimation. In particular, the results suggest that, for the case of five random coefficients, the MACML approach is about 50 times faster than the MSL for the cross-sectional random coefficients case, about 15 times faster than the MSL for the panel inter-individual random coefficients case, and about 350 times or more faster than the MSL for the panel intra- and inter-individual random coefficients case. As the number of alternatives in the unordered-response model increases, one can expect even higher computational efficiency factors for the MACML over the MSL approach. Further, as should be evident in the panel intra- and inter-individual random coefficients case, the MSL is all but practically infeasible when the mixing structure leads to an explosion in the dimensionality of integration in the likelihood function, but these situations are handled with ease in the MACML approach. It is hoped that the MACML procedure will spawn empirical research into rich model specifications within the context of unordered multinomial choice modeling, including autoregressive random coefficients, dynamics in coefficients, space-time effects, and spatial/social interactions.     

基于绿色经济的高速铁路潜在价值研究

Logit模型是一种较为成熟的旅客运输分担率分析方法,它在旅客运输领领域有着广泛的应用。参照绿色经济的定义,首先,本文分析各运输方式的安全性效用、经济性效用、时效性效用、准时性效用、方便性效用、舒适性效用和绿色性效用7个服务特征属性并建立广义效用函数;其次,研究高速铁路客流分担率模型;再次,运用相关数据及最大似然估计法来确定模型的参数;最后,以武汉~广州间各运输方式的客流分担率来研究该模型的应用。     

The estimation of a time-dependent OD trip table with vehicle trajectory samples

Abstract

In this paper we discuss a dynamic origin–destination (OD) estimation problem that has been used for identifying time-dependent travel demand on a road network. Even though a dynamic OD table is an indispensable data input for executing a dynamic traffic assignment, it is difficult to construct using the conventional OD construction method such as the four-step model. For this reason, a direct estimation method based on field traffic data such as link traffic counts has been used. However, the method does not account for a logical relationship between a travel demand pattern and socioeconomic attributes. In addition, the OD estimation method cannot guarantee the reliability of estimated results since the OD estimation problem has a property named the ‘underdetermined problem.’ In order to overcome such a problem, the method developed in this paper makes use of vehicle trajectory samples with link traffic counts. The new method is applied to numerical examples and shows promising capability for identifying a temporal and spatial travel demand pattern.